# 7.  Multiple integral

## Riemann sum

> Very similar to this: [Riemann Sum (univariate calculus)](https://coldmoon.gitbook.io/mathematical-analysis/mathematical-analysis-1/4.-integral#riemann-sums).

#### Intuition define:

Now give you a partition $$P$$:

$$
P: I=\bigcup\_{j=1}^{m} I\_{j}
$$

of this interval with distinguished points $$\xi=\left{\xi\_{1}, \cdots, \xi\_{k}\right}$$.

We defined Riemann sum as:

$$
\sigma(f, P, \xi):=\sum\_{i=1}^{k} f\left(\xi\_{i}\right)\left|I\_{i}\right|
$$

If $$\lim \_{|P| \rightarrow 0} \sigma(f, P, \xi)$$exist and finite, we often denoted it as $$\int\_I f(x)\mathrm d x$$ or :

$$
\int\_{I} f\left(x^{1}, \cdots, x^{n}\right) \mathrm{d} x^{1} \cdots \mathrm{d} x^{n}, \quad \int \cdots \int f\left(x^{1}, \cdots, x^{n}\right) \mathrm{d} x^{1} \cdots \mathrm{d} x^{n}
$$

## Fubini’s Theorem

Let $$X \times Y$$ be an interval in $$\mathbb{R}^{m+n}$$, which is the direct product of intervals $$X \subset \mathbb{R}^{m}$$ and $$Y \subset \mathbb R^{n}$$. If the function $$f: X \times Y \rightarrow \mathbb{R}$$ is integrable over $$X \times Y$$, then all three of the integrals

$$
\int\_{X \times Y} f(x, y) \mathrm{d} x \mathrm{d} y, \quad \int\_{X} \mathrm{d} x \int\_{Y} f(x, y) \mathrm{d} y, \quad \int\_{Y} \mathrm{d} y \int\_{X} f(x, y) \mathrm{d} x
$$

exist and are equal.

## Change of Variable in a Multiple Integral

If $$\varphi: D\_{t} \rightarrow D\_{x}$$ is a diffeomorphism of a bounded open set $$D\_{t} \subset \mathbb{R}^{n}$$ onto a set $$D\_{x}=\varphi\left(D\_{t}\right) \subset \mathbb{R}^{n}$$ of the same type, $$f \in \mathcal{R}\left(D\_{x}\right),$$ and $$\operatorname{supp} f$$ is a compact subset of $$D\_{x}$$, then $$f \circ \varphi\left|\operatorname{det} \varphi^{\prime}\right| \in \mathcal{R}\left(D\_{t}\right)$$, and the following formula holds:

> simplified Chinese: 设 $$\varphi: D\_{t} \rightarrow D\_{x}$$ 为 $$\mathbb{R}^{n}$$ 中的有界开集之间的微分同胚，$$f$$ 为 定义在 $$D\_{x}$$ 上的函数，满足条件 $$\operatorname{supp} f \subset D{x} .$$ 则 $$f \in \mathcal{R}\left(D{x}\right)$$ 当且仅当 $$f \circ \varphi \cdot\left|\operatorname{det} \varphi^{\prime}\right| \in \mathcal{R}\left(D\_{t}\right)$$, 并且如果 $$f \in \mathcal{R}\left(D\_{x}\right)$$，则成立如下的变量代换公式：

$$
\int\_{D\_{x}} f(x) \mathrm{d} x=\int\_{D\_{t}} f \circ \varphi \cdot\left|\operatorname{det} \varphi^{\prime}\right| \mathrm{d} t
$$

## Example

### (1) Polar coordinate transformation

In general polar coordinates $$(r,\theta\_1,\cdots,\theta\_{n−1})$$ in $$\mathbb R^n$$ are introduced via the relations:

$$
\left{\begin{array}{l}
x^{1}\quad=r \cos \theta\_{1} \\
x^{2}\quad=r \sin \theta\_{1} \cos \theta\_{2} \\
\qquad\vdots \\
x^{n-1}=r \sin \theta\_{1} \sin \theta\_{2} \cdot \ldots \sin \theta\_{n-2} \cos \theta\_{n-1} \\
x^{n}\quad=r \sin \theta\_{1} \sin \theta\_{2} \cdot \ldots \cdot \sin \theta\_{n-1} \sin \theta\_{n-1}
\end{array}\right.
$$

because of the orthogonality, we have:

$$
\begin{aligned}
\left|\operatorname{det} \psi^{\prime}\right|^{2} &=\operatorname{det} \psi^{\prime} \cdot \operatorname{det}\left(\psi^{\prime}\right)^{T}=\left|\frac{\partial x}{\partial r}\right|\left|\frac{\partial x}{\partial \theta\_{1}}\right| \cdots\left|\frac{\partial x}{\partial \theta\_{n-1}}\right| \\
&=\left(r^{n-1} \sin ^{n-2} \theta\_{1} \sin ^{n-3} \theta\_{2} \ldots \sin \theta\_{n-2}\right)^{2}
\end{aligned}
$$

so we recall the **Jacobian**:

$$
J=r^{n-1} \sin ^{n-2} \theta\_{1} \sin ^{n-3} \theta\_{2} \cdot \ldots \cdot \sin \theta\_{n-2}
$$

### (2) The volume of  $$n-$$dimensional sphere

Denote the sphere is $$B\_n(r)$$We can easily have the recurrence formula:

$$
V\_{n}(r)=\int\_{-r}^{r} \mathrm{d} x \int\_{B\_{n-1}\left(\sqrt{r^{2}-x^{2}}\right)} \mathrm{d} x^{2} \cdots \mathrm{d} x^{n}=\int\_{-r}^{r} V\_{n-1}\left(\sqrt{r^{2}-x^{2}}\right) \mathrm{d} x
$$

It's trivial that $$V\_1(r)=2r$$, $$V\_2(r)=\pi r^2$$, assume that $$V\_{n}(r)=c\_{n} r^{n}$$, so

$$
\begin{aligned}
V\_{n+1}(r) &=\int\_{-r}^{r} c\_{n}\left(r^{2}-x^{2}\right)^{\frac{n}{2}} \mathrm{d} x=c\_{n} \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} r^{n} \cos ^{n} \varphi \cdot r \cos \varphi \mathrm{d} \varphi \\
&=c\_{n} \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} r^{n+1} \cos ^{n+1} \varphi \mathrm{d} \varphi=c\_{n+1} r^{n+1}
\end{aligned}
$$

with

$$
c\_{n+1}=c\_{n} \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos ^{n+1} \varphi \mathrm{d} \varphi
$$

so that,

$$
V\_{n}(r)=\left{\begin{array}{ll}
2 \frac{(2 \pi)^{k}}{(2 k+1) ! !} r^{2 k+1}, & n=2 k+1 \\
\frac{(2 \pi)^{k}}{(2 k) ! !} r^{2 k}, & n=2 k
\end{array}\right.
$$

$$\blacksquare$$


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